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This paper investigates the class of finitely presented monoids defined byhomogeneous (length-preserving) relations from a computational perspective. Theproperties of admitting a finite complete rewriting system, having finitederivation type, being automatic, and being biautomatic are investigated forthis class of monoids. The first main result shows that for any consistentcombination of these properties and their negations, there is a homogeneousmonoid with exactly this combination of properties. We then introduce the newconcept of abstract Rees-commensurability (an analogue of the notion ofabstract commensurability for groups) in order to extend this result to showthat the same statement holds even if one restricts attention to the class of$n$-ary homogeneous monoids (where every side of every relation has fixedlength $n$). We then introduce a new encoding technique that allows us toextend the result partially to the class of $n$-ary multihomogenous monoids.

On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids